In meteorology or climate, it is common to pool data or consider problems on a region by region basis.


This can make statistical problems more tractable.

National Resource Management Regions

How should assign regions for the analysis of extremes?

Assign regions that are likely to experience similar impacts

Use these regions to inform our statistical analysis

1. Regionalisation

• Clustering
• Dependence of bivariate extremes
• Practicalities
• Classification

2. Visualise spatial dependence

• Max-stable processes

3. Spatial post-processing

Form clusters based on extremal dependence!
(Bernard et al 2013)

• Use only the raw annual maxima
• No information about climate or topography

Use the F-madogram distance
(Cooley et al 2006) \[d(x_i, x_j) = \tfrac{1}{2} \mathbb{E} \left[ \left| F_i(M_{x_i}) - F_j(M_{x_j})) \right| \right]\] where \(M_{x_i}\) is the annual maximum rainfall at location \(x_i \in \mathbb{R}^2\) and \(F_i\) is the distribution function of \(M_{x_i}\).

This distance can be estimated non-parametrically.

For \(M_{x_i}\) and \(M_{x_j}\) with common GEV marginals, \(\theta(x_i - x_j)\) is \[\mathbb{P}\left( M_{x_i} \leq z, M_{x_j} \leq z \right) = \left[\mathbb{P}(M_{x_i}\leq z)\mathbb{P}(M_{x_i}\leq z)) \right]^{\tfrac{1}{2}\theta(x_i - x_j)}. %= \exp\left(\dfrac{-\theta(h)}{z}\right),\]

The range of \(\theta(x_i - x_j)\) is \([1 , 2]\).

Can write our distance measure as a function of the extremal coefficient, \(\theta(x_i - x_j)\), \[d(x_i, x_j) = \dfrac{\theta(x_i - x_j) - 1}{2(\theta(x_i - x_j) + 1)}.\]

Therefore the range of \(d(x_i, x_j)\) is \([0 , 1/6]\).

Partitioning around Medoids (PAM): (Kaufman and Rousseeuw 1990)

1. Randomly select an initial set of \(K\) stations. These are the set of the initial medoids.
2. Assign each station, \(x_i\), to its closest medoid, \(m_k\), based on the F-madogram distance.
3. For each cluster, \(C_k\), update the medoid according to \[m_k = \mathop{\mathrm{argmin}}\limits_{x_i \in C_k} \sum_{x_j \in C_k} d(x_i, x_j).\]
4. Repeat steps 2. – 4. until the medoids are no longer updated.

SWWA EA

Truth

Consider the \(\| x_i - x_j \|\) as the clustering distance.

1. Each station starts in its own cluster
2. For each pair of clusters, \(C_k\) and \(C_k'\), define the distance between the clusters as \[d(C_k, C_{k'}) = \frac{1}{|C_k| |C_{k'}|} \sum_{x_k \in C_k} \sum_{x_{k'} \in C_{k'}} d(x_k, x_{k'}).\]
3. Merge the the clusters with the smallest distance
4. Update the distances relative to the new cluster
5. Repaet steps 3 - 5, until all points are combined in a single cluster

\[d(C_k, C_{k'}) = \frac{1}{|C_k| |C_{k'}|} \sum_{x_k \in C_k} \sum_{x_{k'} \in C_{k'}} d(x_k, x_{k'}).\]

PICTURE

• Classify a station relative to its closest neighbours
• Use a weighted classification \(w\)-kNN

IMAGE

SHINY APP

IMAGE

Where can we assume a common dependence structure for extremes?

Shiny App

SWWA

TAS

Oesting et. al 2017

• approach

• cut the region into two